1. 03/12/13 Trees and CFGs Discrete Structures (CS 173) Derek Hoiem, University of Illinois 1

2. Last class: recursive functions 2

3. Today’s lecture: Trees and CFGs Trees – Examples of uses – Terminology – Induction on trees Context free grammars (CFGs) – What they are, how they work – Induction on CFG 3

4. Tree: special form of graph with “root” and no cycles 4 root

5. Tree terminology Nodes: root, internal, leaf, level, tree height Relations: parent/child/sibling, ancestor/descendant 5 overhead

6. Another example 6 6 4 7 5 2 3 1 root

7. Another example 7 6 4 7 5 2 3 1 root 6 4 7 5 2 3 1

8. Trees for sorting 8 > < > <

9. Decision trees 9

10. Hierarchical data structure 10

11. Trees for clustering 11 b>5 x x x x x x x x x x x 1 23 a b 5 6 a>6 1 yes no yes 2 3

12. More terminology 12 overhead

13. Induction proof on trees 13 overhead

14. Context-free Grammars 14 overhead

15. CFG Example 15 overhead

16. Examples of parse trees 16 Language Fig: Johnson 2007

17. Examples of parse trees 17 Figs: Zhu and Mumford 2007 Scene parse Object Parse

18. Examples of parse trees 18 Stochastic CFG for blackjack actions

19. Induction proof on CFG 19 overhead

20. CFG example 20 * I like to eat apples and bananas overhead

21. Things to remember Trees are a special graph with root and no cycles, with many uses – Sorting, clustering, finding similar values – Decision tree: machine learning, modeling choices – Parse trees: representing hierarchical structures Context free grammars: generate parse trees Proofs on trees: split at root, use inductive hypothesis on subtrees headed by the root’s children 21

22. Next class: more trees Recursion trees and more proofs with trees 22